Aircraft: LARGE TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Taxi
Anomaly: Conflict Ground Conflict; Less Severe; Deviation / Discrepancy - Procedural Clearance; Ground Incursion Runway

Flight was running an hour late into Detroit. Weather at DTW was RVR T/D 1400 Mid 2000 R/O 1600. Shot a CAT III Autoland to 04L. Landing was uneventful. Prior to landing briefed taxi in plan using low vis taxi charts; planned runway exit A7 then V. Turned off as planned on A7. Tower said to turn onto A then V hold short of 04R. This was the first time the Captain (me) had been to DTW. Visibility was extremely limited but safe to continue. I taxied onto A7 and then started to turn right on A instead of left - my first officer immediately corrected me and I corrected the error and turned left on A then right on V. As we approached runway 04R tower cleared us to cross runway 04R then contact ground; they told us there was an aircraft holding in position on 04R but we could not see the traffic. We turned on our exterior lights; crossed runway 04R; extinguished our exterior lights and the first officer contacted ground. Ground Control told us to turn onto taxiway K to the ramp. We were on taxiway V and as we approached taxiway K my first officer said we should turn right on K; I initially thought we were supposed to go left but after the mistake I had already made while taxiing onto taxiway A; I thought I was in error again so turned right onto K. We taxied a short distance on in the fog; to the right I could see bright lights that I didn't make sense to me; I stopped the aircraft looked at the taxi chart again and realized we had taxied across runway 9L/27R. In front of us I could see the dim lights of another aircraft pointing toward us. Ground Control contacted us and told us to hold our position; cleared the other aircraft to begin their taxi and turn right on taxiway U then cleared us to continue forward; turn right on U; right on Y; cross Runway 9L then continue on Y to K5 and to ramp. We followed these instructions and taxied to the ramp without further issue. Runway 09L/27R was not an active runway at the time we crossed it. The visibility was restricted to the point that we could never see any part of the terminal or the terminal lights while taxiing in. We conducted a CAT 3 landing to runway 4L. Landing was uneventful but there was a quick discussion on the unexpected pitch attitude on touchdown. Weather was calm winds; 1/1 temp/dew pt. [VV002] FG RVR1600-2000 reported. Low Vis Arr Taxi Routes were briefed but not in effect. Visibility was restricted due to a combination of fog and bright airport ground lighting. Additionally; taxiway markings were hard to discern. Upon exiting the runway at A7 there was momentary confusion as the lights only led to a right turn south along taxiway A; but instructions were to proceed left to taxiway V and hold short of runway 4R; which we did. Approaching runway 4R we were instructed to cross and contact ground on the other side. There were also instructions given to another aircraft to line up and wait on runway 4R. Being unable to see the departure end of runway 4R; a lot of my attention was on the radio transmissions to be sure that no one was in fact departing runway 4R as we were crossing. This was a distraction. After crossing runway 4R we contacted ground and were given instructions to proceed on taxiway K and to contact North Ramp. I do not recall hearing instructions to turn left on taxiway K. This is a marked hotspot on the taxiway charts. We turned right and immediately found ourselves on runway 9L and stopped. There were no lights warning of the runway boundary. In addition the taxiway/runway markings were very worn and hard to discern. Upon stopping; ground control gave us instructions to proceed on taxiways K; U and Y then to cross runway 9L. Interesting that upon doubling back we did have runway warning lights on taxiway Y/runway 9L; whereas there are none on taxiway K/runway 9L. We proceeded to K5; contacted North Ramp and had no further issues. While taxiing to their gate in low visibility at night; crew made a turn in the wrong direction; crossed a runway; and encountered another aircraft headed in the opposite direction. Ground Control issued revised clearances to both aircraft.